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  • The algebraic closure of the field of real numbers is the field of complex numbers.

  • The algebraic closure of the field of rational numbers is the field of algebraic numbers.

  • The completion of the set of rational numbers is the set of the real numbers, in the sense of either order (Dedekind completion) or metric.

I was wondering if the completion of the set of algebraic numbers in the sense of either order (what kind of order is it?) or metric is the set of complex numbers?

Thanks and regards!

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If you want to use an order relation on algebraic numbers, you'll have to tell us which one, because there is no single "standard" definition of an order on them. –  Robert Israel May 28 '11 at 0:10
    
@Robert: I don't know. I wonder if there is some order on the set of algebraic numbers so that the set can be completed with respect to? –  Tim May 28 '11 at 0:13

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up vote 5 down vote accepted

The metric completion of the algebraics is the complex numbers. It certainly can't be any bigger, and thinking about the completion of the set of all $a+bi$ where $a$ and $b$ are rational shows it can't be any smaller.

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Thanks! Does the set of algebraic numbers have an order to complete with respect to? –  Tim May 28 '11 at 0:11
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@Tim, you can always put an order on any set, but there is no "good" way to order the algebraics, by which I mean, no linear order that respects the addition and multiplication. –  Gerry Myerson May 28 '11 at 0:16

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